Download right triangles-cw - Westminster Public Schools

Algebra / Geometry II:
Unit 6- Right Triangles
SUCCESS CRITERIA:
1. Find the third side of a right triangle using Pythagorean’s Theorem.
2. Determine if a triangle is acute, right, or obtuse using Pythagorean's Theorem.
3. Determine the sides of special right triangles.
4. Find a side or angle of a right triangle using trigonometry.
5. Use trigonometry to answer real world problems.
INSTRUCTOR: Craig Sherman
PMI-NJ Center for Teaching & Learning
Hidden Lake High School
Westminster Public Schools
~1
NJCTL.org
EMPOWER TARGET
MA.10.G.08.04
SCALE THEME
Trigonometric Ratios
PROFICIENCY SCALE:
SCORE
4.0
3.5
3.0
2.0
1.0
REQUIREMENTS
In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that go BEYOND what was taught in
class.
Score 4.0 does not equate to more work but rather a higher level of performance.
In addition to Score 3.0 performance, in-depth inferences and applications with partial success.
The learner exhibits no major errors or omissions regarding any of the information and processes
(simple or complex) that were explicitly taught.
o Find the third side of a right triangle using Pythagorean’s Theorem., AND
o Determine if a triangle is acute, right, or obtuse using Pythagorean's Theorem , AND
o Determine the sides of special right triangles, AND
o Find a side or angle of a right triangle using trigonometry, AND
o Use trigonometry to answer real world problems.
Can do one or more of the following skills / concepts:
There are no major errors or omissions regarding the simpler details and processes as the learner…
o Find the third side of a right triangle using Pythagorean’s Theorem, OR
o Determine if a triangle is acute, right, or obtuse using Pythagorean's Theorem, OR
o Determine the sides of special right triangles, OR
o Find a side or angle of a right triangle using trigonometry, OR
o Use trigonometry to answer real world problems
Know and use the vocabulary
Identify the Basic Elements
With help, a partial understanding of some of the simpler details and process
PMI-NJ Center for Teaching & Learning
~2
NJCTL.org
Similarity in Right Triangles
WORD or CONCEPT
DEFINITION or NOTES
EXAMPLE or
GRAPHIC REPRESENTATION
geometric mean
INSTRUCTION 1: SOPHIA: GEOMETRIC MEAN
INSTRUCTION 2: GEOMETRIC MEAN OF RIGHT TRIANGLE
EXEMPLARS:
GEOMETRIC MEAN with NUMBERS:
GEOMETRIC MEAN with RIGHT TRIANGLES:
GIVEN:
GIVEN:
12 and 16
STEP 1: Proportion
STEP 2: geometric mean
STEP 3: numbers`
STEP 4: cross multiply
STEP 5: solve
STEP 6: square root
=
𝑥
12
=
𝑥
𝑥
=
16
(x)(x) = (12)(16)
x2 = 192
x ≈ 13.8
𝑥
STEP 1: Proportion
STEP 2: geometric mean
=
8
=
8
(altitude)
STEP 3: Parts Hypotenuse
STEP 4: cross multiply
STEP 5: solve
STEP 6: divide
PMI-NJ Center for Teaching & Learning
~3
16
8
=
𝑥
(8)(8) = (16)(x)
64 = 16x
x=4
8
NJCTL.org
Classwork
1. Find the geometric mean of the pairs. Write the answer is simplest radical form.
a. 8 and 32
c. 10 and 15
b. 2 and 24
d. 14 and 18
2. GE is the altitude of triangle DEF. What are the three similar triangles?
Solve for x.
3.
4.
5.
6.
7.
8.
PMI-NJ Center for Teaching & Learning
~4
NJCTL.org
Homework
9. Find the geometric mean of the pairs. Write the answer is simplest radical form.
a. 9 and 25
b. 6 and 54
c. 12 and 20
d. 10 and 28
10. SQ is the altitude of triangle PQR. What are the three similar triangles?
Solve for x.
11.
12.
13.
14.
15.
PMI-NJ Center for Teaching & Learning
NJCTL.org
~1
Pythagorean Theorem and its Converse
WORD or CONCEPT
DEFINITION or NOTES
EXAMPLE or
GRAPHIC REPRESENTATION
leg
hypotenuse
Pythagorean
Theorem
INSTRUCTION 2: SOPHIA TUTORIAL
INSTRUCTION 1: KHAN ACADEMY
Leg2 + Leg2 = Hypotenuse2
a 2 + b2 = c 2
(
Integrated II – Rt. Triangle Trig
~1~
)2 + (
)2 = (
)2
NJCTL.org
Class work
16. In the triangle above, the hypotenuse is side ______, the legs are sides _____ and ______.
Refer the diagram below to answer questions 17-22. Write the answer in simplest radical form.
17. If AB = 9 and BC = 12, then AC = _________.
18. If AB = 11 and AC = 61, then BC = _________.
19. If BC = 45 and AC = 53, then AB = __________.
20. If AC = 9 and AB = 3, then BC = _________.
21. If BC = 14 and AC = 8√7, then AB = __________.
22. If AB = 2√3 and BC = 6√5, then AC = _________.
Classify the sides as lengths of an acute, right, obtuse, or not a triangle.
23. 8, 11, 5
24. 20, 29, 21
25. 6, 2, 2√2
26. 9, 11√2, 7√5
27. Find the perimeter and area of the rectangle.
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Homework
28. In the triangle above the hypotenuse is side ______, the legs are sides _____ and ______.
Refer the diagram below to answer questions 29-34. Write the answer in simplest radical form.
29. If AB = 33 and BC = 56, then AC = _________.
30. If AB = 12 and AC = 37, then BC = _________.
31. If BC = 80 and AC = 89, then AB = __________.
32. If AC = 14 and AB = 5√2, then BC = _________.
33. If BC = 9 and AC = 16, then AB = __________.
34. If AB = 5√2 and BC = 7√6, then AC = _________.
Classify the sides as lengths of an acute, right, obtuse, or not a triangle.
35. 19, 7, 15
36. 9, 4, 3
37. 14, 14, 14√2
38. 3√11, 4√7, 5√6
39. Find the perimeter and area of the triangle.
31cm
29cm
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Special Right Triangles
INSTRUCTION 1: KHAN ACADEMY (30-60-90)
INSTRUCTION 2: SOPHIA TUTORIAL
INSTRUCTION 3: KHAN ACADEMY (45-45-90)
Classwork
40. The length of each leg of an isosceles right triangle is 13 cm. What is the length of the hypotenuse?
41. The length of the hypotenuse of a 45°-45°-90° triangle is 16 cm. What is the length of each leg?
42. The length of the shorter leg of a 30°-60°-90° triangle is 5 cm. What is the length of the longer leg? What is the length of
the hypotenuse?
43. The length of the longer leg of a 30°-60°-90° triangle is 12 cm. What is the length of the hypotenuse? What is the length
of the shorter leg?
44. The length of the hypotenuse of a 30°-60°-90° triangle is 12 cm. What is the length of the longer leg? What is the length
of the shorter leg?
Find the lengths of the missing sides.
45.
46.
18 cm
y
60°
x
60°
x
47.
x
18 cm
y
48.
y
60°
x 45°
y
6 cm
18 cm
49.
x 45° 6 cm
y
50. Find the area of the triangle. Round to the nearest hundredth.
51. A skateboarder constructs a ramp using plywood. The height of the ramp is 4 feet. If the ramp falls at a 60°
angle, what is the length of the plywood
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Homework
52. The length of each leg of an isosceles right triangle is 9 cm. What is the length of the hypotenuse?
53. The length of the hypotenuse of a 45°-45°-90° triangle is 10 cm. What is the length of each leg?
54. The length of the shorter leg of a 30°-60°-90° triangle is 13 cm. What is the length of the longer leg? What is the length
of the hypotenuse?
55. The length of the longer leg of a 30°-60°-90° triangle is 21 cm. What is the length of the hypotenuse? What is the length
of the shorter leg?
56. The length of the hypotenuse of a 30°-60°-90° triangle is 21 cm. What is the length of the longer leg? What is the length
of the shorter leg?
Find the lengths of the missing sides.
57.
24 cm
y
60°
58.
y
x
x
59.
24 cm
60°
60.
x
y 60°
10 cm
y
45°
x
24 cm
61.
x 45° 10 cm
y
62. Find the area of the triangle. Round to the nearest hundredth.
63. A skateboarder constructs a ramp using plywood. The length of the plywood is 7 feet long and falls
at a 40° angle. What is the height of the ramp?
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Trigonometric Ratios
WORD or CONCEPT
DEFINITION or NOTES
EXAMPLE or
GRAPHIC REPRESENTATION
trigonometry
reference angle
side opposite
side adjacent
sine
cosine
tangent
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
INSTRUCTION 1: KHAN ACADEMY INTRODUCTION
INSTRUCTION 2: KHAN BASIC TRIG
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Class work
Refer to the triangle below to answer questions 65-66.
64. side opposite to ∠W is _______side adjacent to ∠W is _______
side opposite to∠B is _______side adjacent to ∠B is _______hypotenuse is _______.
64. sinW = ______
sinB = ______
cosW = ______
tanW = _______.
cosB = ______
tanB = _______.
Evaluate. Round the nearest ten-thousandth.
65. sin90o
67. tan25o
66. cos52o
68. tan88o
69. If the sin30 = .5, the cos60= ______________. Why?
70.
If the cos25= .9063, the sin65=_____________. Why?
Homework
Refer to the triangle below to answer questions 75-76.
71. side opposite to ∠W is _______side adjacent to ∠W is
_______
side opposite to ∠B is _______side adjacent to ∠B is _______hypotenuse is _______.
72. sinW = ______
sinB = ______
cosW = ______
tanW = _______.
cosB = ______
tanB = _______.
Evaluate. Round the nearest ten-thousandth.
73. sin45o
75. cos69o
74. tan 77o
76. tan33o
77. If the sin60 = .8660, the cos30= ______________. Why?
78.
If the cos65= .4226, the sin25=_____________. Why?
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Solving Right Triangles
NOTES:
2-sides
__________ __________ =
3-function 1-ref./
_____ O A
HA
INSTRUCTION 1: SOPHIA TUTORIAL
Class work
Solve the triangle. Round to the nearest hundredth.
79.
BC = ______
80.
STEP 1: Reference ∠
________
270
=
STEP 2: Sides
________ 270
=
𝑥
STEP 3: Function
tan 270 =
STEP 4: Solve
13 * tan 270 =
STEP 5: Solution
𝑥 𝑂
13 𝐴
13
𝑥
6.62
m∠D = ______
81.
m∠I = ______
82.
JL = ______
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Homework
Solve the triangle. Round to the nearest hundredth.
83.
BC = ______
84.
m∠D = ______
63.
m∠I = ______
85.
JL = ______
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Angle of Elevation and Depression
VOCABULARY
WORD or CONCEPT
DEFINITION or NOTES
EXAMPLE or
GRAPHIC REPRESENTATION
angle of elevation
angle of
depression
NOTES:
Angle of Elevation = Angle of Depression
INSTRUCTION 1: SOPHIA TUTORIAL
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Class work
86. Angela flies a kite at a 60o angle of elevation. The kite’s string is 275 feet. Angela’s arm is 4.5 feet off the ground. How
high is the kite off the ground? (Round to the nearest hundredth)
87. An airplane flies 6 miles above the ground. The distance from the start of the runway to the airplane is 15 miles. What
is the angle of depression the airplane must use to land? (Round to the nearest hundredth)
88. You are looking at the top of a tree. The angle of elevation is 78o. The distance from the top of the tree to your position
is 125 feet. If you are 5.25 feet tall, how far are you from the base of the tree? (Round to the nearest hundredth)
Homework
89. An airplane is 20 miles from the start of the runway. The angle of depression the airplane must use to land is 40 o. How
high is the airplane above the ground? (Round to the nearest hundredth)
90. You are standing on a mountain that is 6842 feet high. You look down at your campsite at angle of 38 o. If you are 5.6
feet tall, how far is the base of the mountain from the campsite? (Round to the nearest hundredth)
91. You are looking at the top of a tree. The angle of elevation is 66o. The distance from the top of the tree to your position
is 108 feet. If you are 5.75 feet tall, how tall is the tree? (Round to the nearest hundredth)
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
Right Triangles & Trigonometry
Unit Review
Multiple Choice
- Choose the correct answer for each question. No partial credit will be given.
1.
Find the length of the missing side of the triangle. Write the answer in simplest radical form.
a. 117
2.
15√2
2
c. 18.76
d. √352
b. right
c. obtuse
d. not a triangle
c. 10√3
b. 7.5
d. 9√13
Find the value of x. Write the answer in simplest radical form.
a. 4√2
6.
b. 16√22
Find the value of x. Write the answer in simplest radical form.
a.
5.
d. 3√13
Tell whether the lengths7√2, 3√7, and 5 form the sides of an acute, right, obtuse, or not a triangle.
a. acute
4.
c. √117
Find the length of the missing side of the triangle. Write the answer in simplest radical form.
a. 4√22
3.
b. 9√13
b. 8√2
c.
8√2
2
d. 4
Find the value of x. Write the answer in simplest radical form.
a. 2√2
Integrated II – Rt. Triangle Trig
b. 4√2
~1~
c.
4√2
2
d. 2
NJCTL.org
7.
In the triangle below, what is the side opposite to ∠J.
a. JR
8.
d. LR
b.
1
c.
2
5
7
d.
7
5
Find the m∠V. Round the answer to the nearest hundredth.
a. 33.69o
10.
c. JL
What is tan K?
a. 2
9.
b. LJ
b. 56.31o
c. 41.81o
d. 48.19o
̅̅̅̅. Round the answer to the nearest hundredth.
Find the length of DT
a. 8.31
b. 6.64
Integrated II – Rt. Triangle Trig
c. 6.26
~1~
d. 3.99
NJCTL.org
11.
.In the figure, sin 47 = 15/x, which of the following equations is also true?
x
15
47°
a. sin 43 = x/15
b. cos 43 = 15/x
c. cos 47 = 15/x
d. tan 47 =
x/15
12.
Find the value of x. Write the answer in simplest radical form.
15
30°
x
a.
15√2
c. 10√3
b. 7.5
2
d. 9√13
13. Find the m∠D. Round the answer to the nearest hundredth.
a. 50.71o
b. 39.29o
14. Find the geometric mean of 5 and 9.
a. 3√5
b. 7
c. 35.10o
d. 54.90o
c. 5√3
d. 9√3
c. 5√7
d. 25
15. Solve for x.
16
x
a. 7
9
b. 12
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
16. In the figure, sin 47 = 15/x, which of the following equations is also true?
15
x
47°
a. sin 43 = x/15
b. cos 43 = 15/x
c. cos 47 = 15/x
d. tan 47 = x/15
Short Constructed Response
- Write the answer for each question. No partial credit will be given.
17.
Tell whether the lengths 7√3, 3√11, and 4√5 form the sides of an acute, right, obtuse or not a triangle.
18.
Michael is flying a kite at an angle of 45o. The kite's string is 202 feet long and Amy arm is
4 feet off the ground. How high is the kite off the ground?
19.
Find the perimeter & area of the triangle. Round your answers to the nearest hundredth.
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org
20.
If cos18° = .9511, then what is the sin72°? Explain the relationship between these two values and why the
relationship occurs.
Extended Constructed Response –
Solve the problem, showing all work. Partial credit may be given.
21. You are on the deck of a cruise ship that is 425 m above the water. You look down towards the ocean and
see dolphins. The angle of depression from your line of sight to the dolphins is 42° & 44°, respectively.
a) How far is the first dolphin from the ship?
b) How far is the second dolphin from the ship?
c) What is the distance between the 2 dolphins?
d) If the cruise ship stops at one of its destinations & the dolphins continue to travel toward the ship at a rate of 11
meters per second, how long will it take them to get to the ship?
Integrated II – Rt. Triangle Trig
~1~
NJCTL.org